Question

in the rectangle below, fh = 4x + 2, eg = 5x - 6, and m∠fig = 56°. find ei and m∠ieh.

Explanation:

Step1: Find x using rectangle diagonals property

In a rectangle, diagonals are equal and bisect each other. So \( FH = EG \).
Set \( 4x + 2 = 5x - 6 \).
Subtract \( 4x \) from both sides: \( 2 = x - 6 \).
Add 6 to both sides: \( x = 8 \).

Step2: Calculate length of diagonal

Substitute \( x = 8 \) into \( FH \): \( FH = 4(8)+2 = 32 + 2 = 34 \).
Since diagonals bisect each other, \( EI=\frac{1}{2}EG \) (and \( EG = FH = 34 \)), so \( EI=\frac{34}{2}=17 \).

Step3: Find \( m\angle IEH \)

In rectangle, diagonals bisect each other, so \( EI = HI \), triangle \( IEH \) is isosceles.
\( \angle FIG = 56^\circ \), and \( \angle FIG \) and \( \angle EIH \) are vertical angles, so \( \angle EIH = 56^\circ \).
In \( \triangle IEH \), \( \angle IEH=\frac{180^\circ - 56^\circ}{2}=\frac{124^\circ}{2}=62^\circ \).

Answer:

\( EI = 17 \), \( m\angle IEH = 62^\circ \)